# Restricting a distribution to a non-open subset

If I have an open subset $$U \subset \mathbb{R}^n$$ and a distribution $$\rho \in \mathscr{D}'(U; \mathbb{R})$$, i.e. a continuous linear functional $$\rho: \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{R}) \rightarrow \mathbb{R}$$, then I know that $$\rho$$ can be restricted to an open subset $$V \subset U$$, say to a distribution $$\rho|_V \in \mathscr{D}'(V;\mathbb{R})$$, in the obvious way – we can just extend any test function $$\varphi \in \mathsf{C}_{\mathsf{c}}^\infty(V;\mathbb{R})$$ by zero to get a test function $$\bar{\varphi} \in \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{R})$$, and then set $$\rho|_V(\varphi) := \rho(\bar{\varphi}).$$ My question is, can we do something like this if $$V \subset U$$ isn't open? Since if $$V \subset U$$ is, for example, a submanifold of smaller dimension, then the extension of a nonzero test function by zero is no longer smooth (or even continuous). The specific example I have in mind is this:

Say $$\rho \in \mathscr{D}'(\mathbb{R} \times U;\mathbb{R})$$ is a distribution on $$\mathbb{R} \times U$$, thought of as a time-dependent electric charge density on $$U \subset \mathbb{R}^3$$. Symbolically let's write $$\langle \rho,\psi \rangle_{\mathbb{R} \times U} = \int_{U}\int_\mathbb{R} \rho(t,x)\psi(t,x)\:\mathsf{d}t\mathsf{d}x,$$ where the "function" $$\rho(t,x): \mathbb{R} \times U \rightarrow \mathbb{R}$$ doesn't actually exist unless $$\rho$$ is regular. Then for each time $$t_0 \in \mathbb{R}$$, I should intuitively have a charge density $$\rho_{t_0}$$ on $$U$$, i.e. a distribution $$\rho_{t_0} \in \mathscr{D}'(U; \mathbb{R})$$, obtained by somehow restricting $$\rho$$ to the codimension-$$1$$ submanifold $$\{t_0\} \times U \subset \mathbb{R} \times U$$; this would be given symbolically by $$\langle \rho_{t_0}, \varphi \rangle_U = \int_U \rho(t_0,x)\varphi(x)\:\mathsf{d}x.$$ It seems like this equation can be taken as the definition of $$\rho_{t_0}$$ and makes complete sense, as long as $$\rho$$ is regular. But if $$\rho$$ is an arbitrary distribution, then how can I actually define $$\rho_{t_0}$$? Here are my ideas, which seem to have some issues:

1. Intuitively, it seems like I should define $$\langle \rho_{t_0},\varphi \rangle_U := \langle \rho\delta_{t_0}, \tilde{\varphi} \rangle_{\mathbb{R} \times U}$$, where $$\delta_{t_0} \in \mathscr{D}'(\mathbb{R};\mathbb{R})$$ is the Dirac delta distribution at $$t_0$$ and $$\tilde{\varphi}: \mathbb{R} \times U \rightarrow \mathbb{R}$$ is the function given by $$\tilde{\varphi}(t,x) := \varphi(x)$$. Symbolically we would have $$\langle \rho_{t_0},\varphi \rangle_U = \langle \rho\delta_{t_0}, \tilde{\varphi} \rangle_{\mathbb{R} \times U} = \int_U\int_{\mathbb{R}} \rho(t,x)\delta_{t_0}(t)\tilde{\varphi}(t,x)\:\mathsf{d}t\mathsf{d}x = \int_U \rho(t_0,x)\varphi(x)\:\mathsf{d}x,$$ which is exactly what it should be. The problem is that I need to multiply the two distributions $$\rho$$ and $$\delta_{t_0}$$ together (plus, one is distribution on $$\mathbb{R} \times U$$ and the other is a distribution on $$\mathbb{R}$$).

2. Instead of letting $$\rho$$ be a distribution on $$\mathbb{R} \times U$$, I could take it to be a parametrized family of distributions on $$U$$, say $$\rho: \mathbb{R} \rightarrow \mathscr{D}'(U;\mathbb{R})$$, given by $$t \mapsto \rho_t$$. The problem here is that I no longer have a notion of $$\frac{\partial \rho}{\partial t}$$, and even if I did, then it's hard to make sense of something like $$\frac{\partial^2 \rho}{\partial t \partial x_1}$$.

3. Lastly I could try something like this: to define the value of $$\rho_{t_0}$$ on some test function $$\varphi \in \mathsf{C}_{\mathsf{c}}^\infty(U;\mathbb{R})$$, take some sequence $$\varphi_1,\varphi_2,\dots \in \mathsf{C}_{\mathsf{c}}^\infty(\mathbb{R} \times U;\mathbb{R})$$ of test functions whose supports form a decreasing chain approaching $$\{t_0\} \times U$$, i.e. $$\bigcap_{j=1}^\infty \mathsf{supp}(\varphi_j) = \{t_0\} \times U$$. Then define $$\langle \rho_{t_0},\varphi \rangle_U := \lim_{j \rightarrow \infty} \langle \rho,\varphi_j \rangle_{\mathbb{R} \times U}.$$ This seems kind of complicated, and there may be some issues with well definedness.

Is this notion written down anywhere, in any reference? Any help is really appreciated!

• good question. the canonical approach is 3 which depending on the situation may or may not succeed. What you seem to be interested in is restricting a distribution to an affine subspace. Results in the literature which do that are generically called "trace theorems" (google that). You can also use Hormander's notiion of wavefront set of a distribution. Modulo good geometric position relative to the subspace there are theorems which guarantee that method 3 will work. See Corollary 8.2.7 in Hormander vol 1, second edition. Feb 27, 2020 at 14:39
• That's exactly what I was looking for, thank you so much! I guess I must have missed that in Hormander's book.
– Alec
Feb 27, 2020 at 18:14
• If you are already reading Hormander, then you are on the right track! A remark about trace theorem versus wavefront: if you are only interested in the restriction of one distribution then probably the wavefront theory will be more helpful, if you want to do that in bulk for a lot of distributions at once, you may need a trace thm which typically needs a regularity hypothesis like being in a Sobolev with high enough exponent. See, e.g., these notes I found on the web ltcc.ac.uk/media/london-taught-course-centre/documents/… Feb 27, 2020 at 18:36
• I haven't really read into wavefronts, so I'll take a look at that – thanks again for your suggestions!
– Alec
Feb 27, 2020 at 19:46
• check this out also arxiv.org/abs/1404.1778 Feb 28, 2020 at 0:15